Separating tilting modules (Q1203879)

Let \(A\) be a basic and connected finite dimensional algebra over an algebraically closed field. A tilting module \(_ AT\) (in the sense of \textit{D. Happel} and \textit{C. M. Ringel} [Trans. Am. Math. Soc. 274, 399- 443 (1982; Zbl 0503.16024)]) is called separating if the torsion theory induced by \(T\) in the category \(A\)-mod of finitely generated \(A\)-modules is splitting. By slice is meant slice in the sense of \textit{C. M. Ringel} [``Tame algebras and integral quadratic forms'' (Lect. Notes Math. 1099, 1984; Zbl 0546.16013), p. 180]. The aim of this paper is to show that if \(A\) is a hereditary algebra, then \(_ AT\) is a separating tilting module if and only if the additive category \(\text{add }_ AT\) generated by \(_ A T\) is a slice in \(A\)-mod. Similar results were proven by \textit{Ø. Bakke} [Math. Scand. 63, No. 1, 43-50 (1988; Zbl 0706.16002)].